Calibrating weak rates for the big bang
Kenneth NollettUniversity of South Carolina
andSan Diego State University
Measuring the Neutron LifetimeAmherst Center for Fundamental Interactions
19 September 2014
Precision in astrophysical weak interactions
Today’s theories (& input data) for most astrophysical environments don’t offermuch payoff for high precision weak rates
Rates in many places involve large nuclei, so they’re necessarily either measureddirectly or estimated with a lot of nuclear theory
Theory with percent-level precision (unless I’m missing something) only entersin the Sun and the big bang – simple environments
In the Sun, the uncertainty on the p+p −→ d+e++νe rate is 0.9%, dominatedby two-body physics (in both strong & weak forces)
The amount of helium made in the big bang can be computed to within < 1%,and weak coupling constants from τn are vital to the calculation
Big-bang nucleosynthesis (BBN) as a pillar of cosmology
BBN is the production of the original chemical composition of the universe,during the very hot & dense first ∼ 20 minutes
The composition went from free neutrons & protons to mainly hydrogen &helium, with a little D & Li
BBN yields depend on the universal mean baryon density ρB, so for a long timeBBN was the main handle on ρB
BBN took place at∼ 1 second to 20 minutes, so the light-element yields providea very early window on the universe
In the end, there are only four observables (& perhaps some non-observables)
Ingredients of BBN
1. General relativity
Friedmann-Robertson-Walker metric
ds2 = dt2 − [R(t)]2[
dr2
1− kr2+ r2dΩ2
]describes homogeneous & isotropic universe, sizes scale with R(t)
Insertion into Einstein equations gives the expansion rate(R′(t)
R(t)
)2
=8πG
3ρ
with ρ = ρB + ργ + ρν + ρe + · · ·
In minimal model, densities are assumed homogeneous (doesn’t matter much)
Ingredients of BBN
2. Statistical mechanics of Fermi & Bose gases that fill the universe
ρx =gx
8π3
∫E
exp [(E − µx)/kT ]± 1d3p
Initial conditions are assumed to be equilibrium at a single very high T
Each species (baryons, photons, electrons, 3 neutrino flavors) evolves at awell-defined temperature
T declines during isentropic expansion, since ρx ∝ R−4 for mx kT (γ, ν)and ρx ∝ R−3 for mx & kT
Ingredients of BBN
3. Nuclear cross sections
Abudance evolution proceeds throughnuclear collisions
Cross sections are mainly empirical
Only 12 processes matter∗, enumeratedby Smith, Kawano, Malaney (1993)
on the abundances of helium-3 and deuterium (see fig-ure 1). Then François and Monique Spite at the Obser-vatoire de Paris discovered that certain old stars in ourgalaxy with very thin convective envelopes – rapidly cir-culating regions of a star in which material is well mixed– all contained roughly the same amount of lithium-7.Since spectroscopic measurements show that stars inthis “Spite plateau” contain only very small amounts ofnuclei synthesized in previously existing stars, the starsmust have formed out of nearly primordial gas. Thismeant that the amount of lithium-7 in Spite-plateaustars could be interpreted as the amount of lithium-7synthesized during BBN.
Measurements of light-element abundances con-tinued to advance, and by 2000 they implied a meanbaryon density of 2! 10–31 g cm–3, give or take a factorof three. On the one hand, this was a remarkable case ofdiverse and difficult-to-obtain data all converging tosome value. On the other hand, the formal error barsreflecting known sources of uncertainty had become sosmall that the data points technically disagreed with oneanother. While it was easy to imagine further system-atic errors that could bring the results closer together,due either to the observational techniques or to effectsinvolving the history of the material being observed, itwas much harder to quantify them.
Measurements of deuterium in distant concentra-tions of gas lying between us and even more distantquasars favoured a mean baryon density of about4!10–31 g cm–3, while the simplest interpretation of the
lithium plateau and some of the helium-4 data favouredvalues nearer 1!10–31 g cm–3 (see figure 2). As for theprimordial abundance of helium-3, the post-BBN his-tory of these nuclei is too uncertain to be able to con-strain the mean baryon density. This disagreementprompted a vigorous programme of research by severalgroups in an attempt to improve the measurements andresolve the remaining discrepancies. In the mean time,however, precision cosmological data had started togive BBN a run for its money.
Elemental lightBy the early 2000s, in the midst of the often heateddebate over what to make of the different abundancemeasurements, BBN was no longer the only way todetermine the mean baryon density of the universe. In1992 the COBE satellite revealed that the temperatureof the cosmic microwave background varies by a fewtens of microkelvin on angular scales of 5° or more, thusproviding evidence for density fluctuations in the earlyuniverse that may have seeded cosmic structure. Thenin 2000 the BOOMERANG and MAXIMA experi-ments detected fluctuations on angular scales smallerthan 1°. A key prediction of Big Bang theory, these fluc-tuations are the imprints left by acoustic waves thatpropagated through the plasma just before neutralhydrogen atoms first formed, some 380 000 years afterBBN when the cosmic microwave background wasborn. And since the properties of the plasma dependon the baryon density, the amplitudes of these fluctu-
Big Bang nucleosynthesis (BBN) is a key component of the Big Bangmodel that explains how the light nuclei deuterium, helium-3, helium-4and lithium-7 were created during the first few minutes of the universe. Big Bang theory states that the universe started out some 13.7 billionyears ago in a very hot and dense state that has been expanding andcooling ever since. As described by Einstein’s general theory of relativity,the rate of expansion depends on the amount of mass and energy theuniverse contains. Before BBN took place – when the universe was lessthan 1 s old – matter and energy existed in the form of a hot, dense gas offundamental particles. As the universe cooled, particles with progressivelyless energy populated the universe so that by 1 s only protons, neutronsand lighter stable particles were present. Weak interactions between bothprotons and neutrons and the much lighter electrons, positrons andneutrinos maintained a thermal equilibrium that fixed the relative numbersof neutrons and protons at a certain value. After this, the temperature of
the gas dropped to about 8!109 K, thereby preventing further weakinteractions. From this time onwards, there remained one neutron (n) forevery six protons (i.e. hydrogen nuclei, 1H).
During the next few minutes, nuclei formed. Deuterium nuclei (2H) wereproduced by collisions between protons and neutrons, and further nuclearcollisions led to every neutron grabbing a proton to form the most tightlybound type of light nucleus: helium-4. This process was complete afterabout five minutes, when the universe became too cold for nuclearreactions to continue. Tiny amounts of deuterium, helium-3 and beryllium-7 were produced as by-products, with the latter undergoing beta decay toform lithium-7. Almost all of the protons that were not incorporated intohelium-4 nuclei remained as free particles, and this is why the universe isclose to 25% helium and 75% hydrogen by mass everywhere we look. Theother nuclei are less abundant by several orders of magnitude.
By measuring the intensity of atomic spectral lines in astrophysicalobjects, astronomers can infer the number of nuclei of a given type perhydrogen nucleus. These nuclear abundances produced during BBNdepend on the density of matter (or baryon density) during those first fewminutes, which can be related directly to the baryon density we see today.Any effect that changes the early thermal evolution of the universe or theinteractions between the nuclei would also leave traces in the abundances,which means BBN provides an important probe of the early universe.
If we assume that only the particles and forces contained in the StandardModel of particle physics were present during BBN, then the baryon densitymeasured by NASA’s WMAP mission (and corroborated by the deuteriumabundance) determines the initial chemical composition of the universe:mostly hydrogen, with roughly 0.08 helium-4 atoms, 10–5 deuterium atoms,10–5 helium-3 atoms and 10–10 lithium atoms per hydrogen atom, but nodetectable amount of anything else. All the other elements in the cosmoswere synthesized much later inside stars or in cosmic-ray collisions.
How Big Bang nucleosynthesis works
1
2
3
4
5
6
7
8
9
10
11
12
2H + 1H2H + 2H2H + 2H2H + 3H3H + 4He3He + n
1H + n
3He + 2H3He + 4He7Li + 1H7Be + n
2H + 3He + 3He + n3H + 1H4He + n
7Li + 3H + 1H4He + 1H
7Be + 4He + 4He7Li + 1H
n 1H + e– +
1H
1
23 4
5
67
9
8
11
10
12
2H
n
3He
7Be
7Li
4He
3H
physicsworld.comFeature: Big Bang nucleosynthesis
22 Physics World August 2007
Calculations with huge reaction networks and nuclei to CNO region have beendone
Weak p + l ↔ n + l′ rates are all normalized to neutron lifetime & computedfrom weak-interaction physics
BBN in three easy steps
At temperatures above T ∼ 1010 K, the ratio of neutrons to protons is governedby equilibrium enforced by weak interactions:
νe + n←→ p+ e−
and “crossed” diagrams
Nucleosynthesis starts at T ∼ 1010 K, when the rates for processes maintainingequilibrium become slower than the universal expansion: Γn↔p < R′/R
The neutron/proton ratio freezes out at
nn
np= exp[−(mn −mp)/kT ] ∼
1
7
This is Weak Freezeout
Some destruction of neutrons by e+ + n→ p+ νe and νe + n→ p+ e− andfree decay follows, but it doesn’t have much time
BBN in three easy steps
At the time of weak freezeout, relative amounts of light nuclei are in NuclearStatistical Equilibrium (NSE)
Almost all nucleons are free, small amountsof D, 3He, 3H, and 4He
Dropping T gradually favors A = 3 and 4
At ∼ 5 minutes, almost all neutrons are in4He (large per-particle binding energy)
on the abundances of helium-3 and deuterium (see fig-ure 1). Then François and Monique Spite at the Obser-vatoire de Paris discovered that certain old stars in ourgalaxy with very thin convective envelopes – rapidly cir-culating regions of a star in which material is well mixed– all contained roughly the same amount of lithium-7.Since spectroscopic measurements show that stars inthis “Spite plateau” contain only very small amounts ofnuclei synthesized in previously existing stars, the starsmust have formed out of nearly primordial gas. Thismeant that the amount of lithium-7 in Spite-plateaustars could be interpreted as the amount of lithium-7synthesized during BBN.
Measurements of light-element abundances con-tinued to advance, and by 2000 they implied a meanbaryon density of 2! 10–31 g cm–3, give or take a factorof three. On the one hand, this was a remarkable case ofdiverse and difficult-to-obtain data all converging tosome value. On the other hand, the formal error barsreflecting known sources of uncertainty had become sosmall that the data points technically disagreed with oneanother. While it was easy to imagine further system-atic errors that could bring the results closer together,due either to the observational techniques or to effectsinvolving the history of the material being observed, itwas much harder to quantify them.
Measurements of deuterium in distant concentra-tions of gas lying between us and even more distantquasars favoured a mean baryon density of about4!10–31 g cm–3, while the simplest interpretation of the
lithium plateau and some of the helium-4 data favouredvalues nearer 1!10–31 g cm–3 (see figure 2). As for theprimordial abundance of helium-3, the post-BBN his-tory of these nuclei is too uncertain to be able to con-strain the mean baryon density. This disagreementprompted a vigorous programme of research by severalgroups in an attempt to improve the measurements andresolve the remaining discrepancies. In the mean time,however, precision cosmological data had started togive BBN a run for its money.
Elemental lightBy the early 2000s, in the midst of the often heateddebate over what to make of the different abundancemeasurements, BBN was no longer the only way todetermine the mean baryon density of the universe. In1992 the COBE satellite revealed that the temperatureof the cosmic microwave background varies by a fewtens of microkelvin on angular scales of 5° or more, thusproviding evidence for density fluctuations in the earlyuniverse that may have seeded cosmic structure. Thenin 2000 the BOOMERANG and MAXIMA experi-ments detected fluctuations on angular scales smallerthan 1°. A key prediction of Big Bang theory, these fluc-tuations are the imprints left by acoustic waves thatpropagated through the plasma just before neutralhydrogen atoms first formed, some 380 000 years afterBBN when the cosmic microwave background wasborn. And since the properties of the plasma dependon the baryon density, the amplitudes of these fluctu-
Big Bang nucleosynthesis (BBN) is a key component of the Big Bangmodel that explains how the light nuclei deuterium, helium-3, helium-4and lithium-7 were created during the first few minutes of the universe. Big Bang theory states that the universe started out some 13.7 billionyears ago in a very hot and dense state that has been expanding andcooling ever since. As described by Einstein’s general theory of relativity,the rate of expansion depends on the amount of mass and energy theuniverse contains. Before BBN took place – when the universe was lessthan 1 s old – matter and energy existed in the form of a hot, dense gas offundamental particles. As the universe cooled, particles with progressivelyless energy populated the universe so that by 1 s only protons, neutronsand lighter stable particles were present. Weak interactions between bothprotons and neutrons and the much lighter electrons, positrons andneutrinos maintained a thermal equilibrium that fixed the relative numbersof neutrons and protons at a certain value. After this, the temperature of
the gas dropped to about 8!109 K, thereby preventing further weakinteractions. From this time onwards, there remained one neutron (n) forevery six protons (i.e. hydrogen nuclei, 1H).
During the next few minutes, nuclei formed. Deuterium nuclei (2H) wereproduced by collisions between protons and neutrons, and further nuclearcollisions led to every neutron grabbing a proton to form the most tightlybound type of light nucleus: helium-4. This process was complete afterabout five minutes, when the universe became too cold for nuclearreactions to continue. Tiny amounts of deuterium, helium-3 and beryllium-7 were produced as by-products, with the latter undergoing beta decay toform lithium-7. Almost all of the protons that were not incorporated intohelium-4 nuclei remained as free particles, and this is why the universe isclose to 25% helium and 75% hydrogen by mass everywhere we look. Theother nuclei are less abundant by several orders of magnitude.
By measuring the intensity of atomic spectral lines in astrophysicalobjects, astronomers can infer the number of nuclei of a given type perhydrogen nucleus. These nuclear abundances produced during BBNdepend on the density of matter (or baryon density) during those first fewminutes, which can be related directly to the baryon density we see today.Any effect that changes the early thermal evolution of the universe or theinteractions between the nuclei would also leave traces in the abundances,which means BBN provides an important probe of the early universe.
If we assume that only the particles and forces contained in the StandardModel of particle physics were present during BBN, then the baryon densitymeasured by NASA’s WMAP mission (and corroborated by the deuteriumabundance) determines the initial chemical composition of the universe:mostly hydrogen, with roughly 0.08 helium-4 atoms, 10–5 deuterium atoms,10–5 helium-3 atoms and 10–10 lithium atoms per hydrogen atom, but nodetectable amount of anything else. All the other elements in the cosmoswere synthesized much later inside stars or in cosmic-ray collisions.
How Big Bang nucleosynthesis works
1
2
3
4
5
6
7
8
9
10
11
12
2H + 1H2H + 2H2H + 2H2H + 3H3H + 4He3He + n
1H + n
3He + 2H3He + 4He7Li + 1H7Be + n
2H + 3He + 3He + n3H + 1H4He + n
7Li + 3H + 1H4He + 1H
7Be + 4He + 4He7Li + 1H
n 1H + e– +
1H
1
23 4
5
67
9
8
11
10
12
2H
n
3He
7Be
7Li
4He
3H
physicsworld.comFeature: Big Bang nucleosynthesis
22 Physics World August 2007
Low ρ and T , Coulomb barriers, disappearance of neutrons, fragility to protonreactions, and lack of stable A = 5,8 nuclei all cause Final Freezeout
BBN in a nutshell
1. Weak Freezeout(∼ 1 second)
2. Statisticalequilibrium &quasi-equilibrium(∼ 1 secondto 5 minutes)
3. Final Freezeout(> 5 minutes)
10!1100101102
Temperature (109 K)
10!24
10!19
10!14
10!9
10!4
101
Mas
s Fra
ctio
n
np7Li,7BeD4He3H,3He6Li
1/60 1 5 15 60Minutes:
The “Schramm plot”
Yields depend on one variable, nB/nγ
Conventional units are ΩB ≡ ρB/ρcrit
ΩBh2 =
8πGρB/(3× 104 km2 s−2 Mpc−2)
h ∼ 0.7 is Hubble’s constant incustomary units, so h2 ∼ 1/2
Widths of curves reflect nuclear inputs(More on this in a few minutes...)
Need to find matter that has not beenprocessed post-BBN & compare
BBN today
The Big Question is now
Are the primordial abundances consistentwith the standard cosmology?
The only ΛCDM parameter that BBNdepends on is ΩBh
2 ∝ nB/nγ
With 1.3% precise ΩBh2 from CMB, BBN
gives very precise predictions
If the answer is “no,” there are interestingthings to be learned about:
neutrinos model atmospheresgravity stellar evolutionall of the above none of the above
...but we can’t tell a priori which one(s)
Standard BBN as a precise theory
Deuterium nuclear inputs have improved considerably in the last decade, nowdominated by d+ p −→ 3He + γ
D/H = (2.51± 0.08)× 10−5 (2.5% nuclear, 2% ΩBh2)
Primordial 3He is not yet observable; it depends on much of the same nucleardata & is kind of flat in ΩBh
2
3He/H = (1.07± 0.04)× 10−5, mostly nuclear
A major logjam in 3He + α −→ 7Be + γ precision broke in the ’00s
Li/H = (5.5± 0.4)× 10−10, . 2% from ΩBh2
(Li probably could be handled better – long story)
BBN post-WMAP: Precise 4He predictions
Convention is to consider Primordial He mass fraction YP
This is not my fault – observation & theory give nHe/nH more naturally
At the end of BBN, all but ∼ 10−5 of neutrons are in 4He(YP specifies the isospin density of the universe)
YP thus probes weak-interaction freezeout at ∼ 1 second, insensitive to ΩB
The ratio of weak rates to the expansion rate at ∼ 1 s determines the freezeouttemperature & therefore YP
Neutron “decay” in BBN
Weak rates are all∝ G2V +3G2
A, matched to τn at the start of a BBN calculation
SupposedlyGV &GA/GV are now known to a precision equivalent to ∆τn ∼ 2 s– there’s not much history of using them instead
The (shallow) dependence of YP onΩBh
2 is mainly through neutrondestruction after freezeout
But even that includes interactionswith the lepton gases
Blue: free decay
Red: n+ νe −→ p+ e−
Green: n+ e+ −→ p+ νe
Others: n production
!"en !ln2
hftiðmec2Þ5
Z 1
#mnp
F½Z; Ee%ðEe & #mnpÞ2
' EeðE2e &mec
2Þ1=2½S"e%½1& Se&%dEe; (12)
!ndecay !ln2
hftiðmec2Þ5
Z #mnp
mec2F½Z; Ee%ð#mnp & EeÞ2
' EeðE2e &mec
2Þ1=2½1& S !"e%½1& Se&%dEe; (13)
!pe& !"e! ln2
hftiðmec2Þ5
Z #mnp
mec2F½Z; Ee%ð#mnp & EeÞ2
' EeðE2e &mec
2Þ1=2½S !"e%½Se&%dEe; (14)
where Ee is the total electron or positron energy as appro-priate, mec
2 is the electron rest mass, and F½Z; Ee% is theCoulomb correction Fermi factor which will be discussedin detail below. Note that the nuclear charge relevant hereis Z ¼ 1. Se&=þ and S"e= !"e
are the phase space occupationprobabilities for electrons/positrons and neutrinos/antineu-trinos, respectively. For neutrinos and electrons with en-ergy distributions with the expected thermal form, theoccupation probabilities are
S"e¼ 1
eE"=T"&$"e þ 1; (15)
S !"e¼ 1
eE"=T"&$ !"e þ 1; (16)
Se ¼1
eEe=T þ 1; (17)
where T" is the neutrino temperature parameter, $" is theneutrino degeneracy parameter (the ratio of chemical po-tential to temperature), and E" is the appropriate neutrinoor antineutrino energy. In what follows, we have neglectedeþ=& annihilation corrections to the weak decoupling pro-cess [22] and the associated neutrino spectral distortion.
We take
ln2
hfti ¼ cðmec2Þ5@c * # * G
2FjCV j2ð1þ 3jCA=CV j2Þ
2%3 ; (18)
where GF ! 1:166' 10&11 MeV&2 is the Fermi constant,CV and CA are the vector and axial vector coupling con-stants, respectively, and we have taken the absolute squaresof the Fermi and Gamow-Teller matrix elements for thefree nucleons to be jMFj2 ¼ 1 and jMGTj2 ¼ 3, respec-tively. Here, # is a factor which includes both Coulomb andother (‘‘radiative correction’’) effects which amount to afew percent change in the effective ft value, hfti.
Of course, CV and CA are coupling constants that arerenormalized by the particular strong interaction environ-ment characterizing free neutrons and protons. (Absentstrong interactions CV ¼ CA ¼ 1.) Given that these area priori unknowns, as is #, we follow the standard proce-
dure [15]: we take the free neutron decay rate as theproduct of Eq. (18) and the phase space factor inEq. (13) (with S !"e
¼ Se& ¼ 0), and we then set this equalto the inverse of the laboratory-measured free neutronlifetime, &n. The world average of the laboratory measure-ments is &n ¼ 885:7 seconds [23].Note that changing the prescription for the Coulomb
correction factor F½Z; Ee% in Eq. (13) will have the effectof renormalizing the effective free nucleon weak interac-tion matrix elements (i.e., renormalizing hfti) for a given&n. As we will see below, this renormalization will be thedominant component of the Coulomb correction alterationin, e.g., the 4He BBN yield.The rates for all the individual weak reactions are shown
as functions of temperature in Fig. 2. At high temperaturesthe forward and reverse rates of the lepton capture reac-tions in Eq. (1) and (2) dominate the neutron-proton inter-conversion process. Note that the rates for the forwardprocess in Eq. (2) and the reverse process in Eq. (1) areaffected by the threshold, #mnp þmec
2. At lower tem-peratures, this threshold makes these rates relatively slowerthan the rates for the lepton capture channels without thisthreshold, i.e., the forward process in Eq. (1) and thereverse process in Eq. (2).This figure shows that at a lower temperature (T +
#mnp), the electron capture rate !e&p and the three-bodyrate !pe& !"e
track each other closely, differing by a factor oforder unity. This is readily explained as follows. First, notethat the integrands in the phase space factors in Eqs. (9) and
1e-25
1e-20
1e-15
1e-10
1e-05
1
100000
0.001 0.01 0.1 1 10
wea
k ra
te (
s-1)
Temperature(MeV)
FIG. 2 (color online). All six weak reaction rates as a functionof temperature. The solid (red) line is for !"en, the dashed(green) line is for !eþn, the dotted (blue) line is for !ndecay , the
small-dashed (pink) line is for ! !"ep, the dash-dotted (cyan) lineis for !e&p, and the black dotted-spaced line is for !pe& !"e
. Alllepton chemical potentials are set to zero here.
WEAK INTERACTION RATE COULOMB CORRECTIONS IN . . . PHYSICAL REVIEW D 81, 065027 (2010)
065027-3
Smith & Fuller, PRD 81, 065027 (2010)
BBN post-WMAP: Precise 4He predictions
YP cares a lot about fine details of weak rates and early thermal conditions
was found to be insensitive to in the range, 1010109. Dicus et al. 14 attempted to calculate the thermo-dynamic corrections, and found YP /YP0.04% , butonly included the effect of the electron mass on the weakrates. Heckler estimated the effect on YP and foundYP /YP0.06% . It should be noted that his value forthe change in neutrino temperature was incorrect. In anyevent, the thermodynamic correction to YP is small.
B. Incomplete neutrino decoupling
The standard code assumes that neutrinos decoupled com-pletely before e annihilations. It has been pointed out thatthis assumption is not strictly valid 14. Neutrinos are‘‘slightly coupled’’ when e pairs are annihilated, and henceshare somewhat in the heat released. The first calculations14,46,47 of this effect were ‘‘one-zone’’ estimates thatevolved integrated quantities through the process of neutrinodecoupling. More refined ‘‘multi-zone’’ calculations trackedmany energy bins, assumed Boltzmann statistics and madeother approximations 25,48. The latest refinements haveincluded these small effects as well 49–51. Fields et al.52 incorporated the slight effect of the heating of neutrinosby e annihilations into the standard code and found a shiftin 4He production, YP1.5104, which is insensitiveto for 1010109.
V. SUMMARY
All of the physics corrections we investigated have beenstudied elsewhere. However, not all of them have beenimplemented in a full code; some have been implementedincorrectly; and there have been changes in some of thephysics corrections. Further, the issue of numerical accuracyof the standard code has not been comprehensively and co-herently addressed. Finally, the corrections have been imple-mented in a patchwork fashion, so that the users of manycodes do not know which corrections are in, which are out,and which may be double counted e.g., by adding the nu-merical correction and running a small step size. As notedearlier results of a number of BBN codes gave a 1% spreadin the prediction for YP with the same value of and n .The goal of this work was a calculation of the primordial
4He abundance to a precision limited by the uncertainty inthe neutron mean lifetime, n2sec, or YP /YP0.2% , with reliable estimates of the theoretical error. Toachieve this goal we created a new BBN code, designed,engineered and tested to this numerical accuracy. To thisbaseline code we added the microphysics necessary toachieve our accuracy goal – Coulomb and zero-temperatureradiative corrections, finite-nucleon-mass corrections, finite-temperature radiative corrections, QED thermodynamicalcorrections, and the slight heating of neutrinos by e anni-hilations. These corrections—coincidentally all positive—increase the predicted 4He abundance by YP0.0049 or2% . Table V summarizes these corrections for 51010. For each physical or numerical effect, we havebeen careful to control the error in YP introduced by approxi-mations or inaccuracies to be well below 0.1% . With confi-dence we can state that the total theoretical uncertainty is lessthan 0.1% .Summarizing our work in one number
YP510100.24620.0004expt
0.0002 theory. 42
Further, the precise value of the baryon density inferredfrom the Burles-Tytler determination of primordial D abun-dance, Bh20.0190.001 40,54, leads to the pre-diction: YP0.24640.0004 (expt) 0.0005 (D/H) 0.0002 theory.
TABLE V. Summary of results for 5.01010. By baseline we mean the results of our BBN codewithout any of the physics effects listed, and with small numerical errors see Fig. 1.
Cumulative Effect AloneYP YP(104) YP /YP(% ) YP(104) YP /YP(% )
Baseline 0.2414Coulomb and T0 radiative 0.2445 31 1.28 31 1.28finite mass 0.2457 43 1.78 12 0.50finite T radiative 0.2460 46 1.90 3 0.12QED plasma 0.2461 47 1.94 1 0.04residual -heating 0.2462 49 2.00 1.5 0.06
FIG. 17. Relative finite-temperature QED change in the neutrinotemperature, as a function of photon temperature. Note that thezero-temperature limit is altered from the standard value by about0.08% .
ROBERT E. LOPEZ AND MICHAEL S. TURNER PHYSICAL REVIEW D 59 103502
103502-12 Lopez & Turner 1999
Lopez & Turner (1999) computed YP with an error budget of ∆YP = 0.0002
Olive, Steigman, & Walker (2000) agree to ∆YP = 0.0001
Mangano & collaborators (more independent) agree to ∆YP = 0.0004
The Mangano code is coming into wide use & the issue is in danger of beinglost (Lopez now does high-frequency trading)
BBN post-WMAP: Precise 4He predictions
The neutron lifetime is a big part of the (small) error budget
Source τn ∆YPPDG 2004–10 885.7± 0.8 sec 0.00016Serebrov 2005 878.5± 1.0 0.00020Pichlmaier 2010 880.7± 2.5 0.00050PDG 2012 880.1± 1.1 0.00022PDG 2014 880.3± 1.1 0.00022
Total spread across the table is ∆YP = 0.0015
Planck gives ΩBh2 = 0.02214±0.00024, robust against varying assumptions
dYP/d(ΩBh2) = 0.43 so ∆YP = 0.00010 from ΩBh
2
In sum, YP = 0.2471 ± 0.0002(theory) ± 0.0002(τn) ± 0.0001(CMB)(using 2014 PDG)
So YP is an astronomical quantity predicted to < 0.5% – unique outside orbitalmechanics?
Helium: Percent compositions from 70 Mpc away?
He/H is inferred from nebular emission inblue compact dwarf galaxies (BCD)
Peimbert et al. 2007 study 5 objects insome detail, 0.2477± 0.0029
Izotov & Thuan (2013) study 111 objects,0.254± 0.003 (August 2014 paperI haven’t digested has0.2551± 0.0022)
Aver, Olive, Skillman have explored errorestimation for subsets of Izotov,currently 0.2535± 0.0036†
YBBN = 0.2471± 0.0005No. 1, 2010 THE PRIMORDIAL ABUNDANCE OF 4He L69
(a) (b)
Figure 1. Linear regressions of the helium mass fraction Y vs. oxygen abundance for H ii regions in the HeBCD sample. The Ys are derived with the He i emissivitiesfrom Porter et al. (2005). The electron temperature Te(He+) is varied in the range (0.95–1) × Te(O iii). The oxygen abundance is derived adopting an electrontemperature equal to Te(He+) in (a) and to Te(O iii) in (b).
7. The equivalent width of the He i λ4471 absorptionline is chosen to be EWabs(λ4471) = 0.4 Å, follow-ing Izotov et al. (2007) and Gonzalez Delgado et al.(2005). The equivalent widths of the other absorptionlines are fixed according to the ratios EWabs(λ3889)/EWabs(λ4471) = 1.0, EWabs(λ5876)/EWabs(λ4471) =0.8, EWabs(λ6678)/EWabs(λ4471) = 0.4 and EWabs(λ7065)/EWabs(λ4471) = 0.4. The EWabs(λ5876)/EWabs(λ4471) and EWabs(λ6678)/EWabs(λ4471) ratioswere set equal to the values predicted for these ratiosby a Starburst99 (Leitherer et al. 1999) instantaneousburst model with an age of 3–4 Myr and a heavy ele-ment mass fraction Z = 0.001–0.004, 0.8 and 0.4, re-spectively. These values are significantly higher than thecorresponding ratios of 0.3 and 0.1 adopted by Izotovet al. (2007). We note that the value chosen for theEWabs(λ5876)/EWabs(λ4471) ratio is also consistent withthe one given by Gonzalez Delgado et al. (2005). Since theoutput high-resolution spectra in Starburst99 are calculatedonly for wavelengths ! 7000 Å, we do not have a predictionfor the EWabs(λ7065)/EWabs(λ4471) ratio. We set it to beequal to 0.4, the value of the EWabs(λ6678)/EWabs(λ4471)ratio.
8. The He ionization correction factor ICF(He++He++) isadopted from Izotov et al. (2007).
3. THE PRIMORDIAL He MASS FRACTION Yp AND THESLOPE dY/dZ
Two Y–O/H linear regressions for the HeBCD galaxy sampleof Izotov et al. (2007), with the above set of parameters, areshown in Figure 1. The two regression lines differ in the wayoxygen abundances have been calculated. For the first regressionline (Figure 1(a)), oxygen abundances have been derived bysetting the temperature of the O++ zone equal to Te(He+),while for the second (Figure 1(b)), they have been derivedby adopting the temperature Te(O iii) derived from the [O iii]λ4363/(λ4959+λ5007) line flux ratio.
The primordial values obtained from the two regressions inFigure 1, Yp = 0.2565 ± 0.0010 and Yp = 0.2560 ± 0.0011,are very similar but are significantly higher than the value Yp =0.2516 ± 0.0011 obtained by Izotov et al. (2007) for the same
galaxy sample. The 2% difference is due to the inclusion of thecorrection for fluorescent excitation of H lines, the correctionfor a larger correction for collisional excitation to the Hβ flux,and larger adopted equivalent widths of the stellar He i 5876,6678, and 7065 absorption lines. We adopt the value of Yp fromFigure 1(a), where both O/H and Y are calculated with the sametemperature Te = Te(He+).
We have varied the ranges of some parameters to study howthe value of Yp is affected by these variations. We have foundthat varying the fraction of fluorescent excitation of the hydrogenlines between 0% and 2%, and/or setting Te(He+) = Te(O iii)or changing Te(He+) in the range (0.9–1.0)× Te(O iii) (insteadof making it change between 0.95 and 1.0 × Te(O iii)), result ina change of Yp between 0.254 and 0.258. Additionally, addinga systematic error of 1% caused by uncertainties in the He iemissivities (Porter et al. 2009) gives Yp = 0.2565 ± 0.0010(stat.) ± 0.0050 (syst.), where “stat” and “syst” refer to statisticaland systematic errors, respectively. Thus, the value of Yp derivedin this Letter is 3.3% greater than the value of 0.2482 obtainedfrom the three-year WMAP data, assuming SBBN (Spergel et al.2007). However, it is consistent with the Yp = 0.25+0.10
−0.07 obtainedby Ichikawa et al. (2008) from the available WMAP, ACBAR,CBI, and BOOMERANG data (actually, the peak value intheir one-dimensional marginalized distribution of Yp (theirFigure 3) is equal to 0.254).
Using Equation (3), we derive from the Y – O/H linearregression (Figure 1(a)) the slopes dY/dO = 2.46 ± 0.45(stat.)and dY/dZ = 1.62 ± 0.29(stat.). These slopes are shallower thanthe ones of 4.33 ± 0.75 and 2.85 ± 0.49 derived by Izotov et al.(2007).
4. DEVIATIONS FROM SBBN
We now use our derived value of the primordial He abundancealong with the observed primordial abundances of other lightelements to check the consistency of SBBN. Deviations fromthe standard rate of Hubble expansion in the early universe canbe caused by an extra contribution to the total energy density, forexample, by additional flavors of neutrinos. The total number ofdifferent species of weakly interacting light relativistic particlescan be conveniently be parameterized by Nν , the “effectivenumber of light neutrino species.”
Izotov & Thuan 2010
Errors as small as 0.0015 have been claimed in the past; underlying atomicdata may have problems amounting to ∆YP ∼ 0.005
Changes in atomic data shifted everyone up ∆YP ∼ 0.010 a few years ago
A timely example: BBN from a neutrino’s point of view
BBN has a long history of constrainingneutrino-like species using the sensitivityat 1 second
Each (doublet) ν species carries ∼ 15% ofenergy density during BBN
−→ the sum sets expansion timescales
More neutrinos −→ faster expansion−→ weak freezeout at higher T−→ more neutrons −→ higher YP
Since YP also depends (weakly) on ΩBh2,
another input is neededNeff = 0 to 10 shown
Counting neutr(on|ino)s using helium
We can use ΩBh2 from CMB + assumption of unchanging nB/nγ after BBN
Or we can fit YP jointly with D/H (assumes less)
This program has received new interest now that the CMB probes the expansionrate at the time of CMB formation
Cosmologists tend to measure the expansion rate as an equivalent number ofthermally-populated neutrino species
Neff = 3.046 in the standard model (after small corrections)
Neutrino counting with BBN & the CMB
A couple of years ago, there were hints from the CMB that Neff ∼ 3.8± 0.4
Now we have:
The data agree, but together they like neither Nν = 3 nor Nν = 4
Salmon: CMB only Blue: BBN only Green: combinedjunk update of Nollett & Steigman, arXiv:1312.5725
Neff = 3.30± 0.27 (CMB), Neff = 3.56± 0.23 (BBN), Neff = 3.40± 0.16 (joint)
Yes, noninteger Neff is meaningful – e.g. light scalar particles
Comparison of τn with what we’re trying to do
At fixed ΩBh2, one additional neutrino species produces ∆YP ' 0.013
An additional second of neutron lifetime produces ∆YP ' 0.00021
The full difference between the “old” PDG lifetime & the Serebrov lifetime is∆YP = 0.0015 (from ∆τn = 6.8 s)
So the τn spread gives ∆Neff ∼ 0.0015/0.013 ∼ 0.12
By comparison, the CMB is unlikely to measure Neff to within much better than∆Neff ∼ 0.20
This all compares with reasonable observational errors today of ∆YP ∼ 0.005
The same information, graphically
Here are abundances as functions of Neff
(ΩBh2 slightly outdated)
Pink band in YP shows errors around 2011PDG recommended τn
Black lines on either side are 2004-2010PDG & Serebrov
(Black lines in lower panels reflect othernuclear uncertainties)
Nollett & Holder (2012), partial update
What I would like to see
The best thing for me would be an agreed τn with an error of ∼ 1 s (again)
BBN has intrinsic interest as a source of very precise predictions arising fromthe standard cosmology, probing very early times
Even if astronomers can’t match the theory’s precision now, it’s good to havethe target out there (0.2% prediction!)
Any problem with τn sits below my predictions & skews my conclusions by∼ σ/2
YP also feeds into modeling of CMB anisotropies (which currently constrain YPby ±0.06!)
I’m not sure they’ll ever be sensitive at the percent level, though
